# Fungrim entry: 944a14

$R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt$
Assumptions:$x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)$
TeX:
R_D\!\left(x, y, z\right) = \frac{3}{2} \int_{0}^{\infty} \frac{1}{\left(t + x\right) \left(t + y\right) {\left(t + z\right)}^{3 / 2}} \, dt

x \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left(-\infty, 0\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; y \ne 0\right)
Definitions:
Fungrim symbol Notation Short description
CarlsonRD$R_D\!\left(x, y, z\right)$ Degenerate Carlson symmetric elliptic integral of the third kind
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
OpenInterval$\left(a, b\right)$ Open interval
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("944a14"),
Formula(Equal(CarlsonRD(x, y, z), Mul(Div(3, 2), Integral(Div(1, Mul(Mul(Add(t, x), Add(t, y)), Pow(Add(t, z), Div(3, 2)))), For(t, 0, Infinity))))),
Variables(x, y, z),
Assumptions(And(Element(x, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(y, SetMinus(CC, OpenInterval(Neg(Infinity), 0))), Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Or(NotEqual(x, 0), NotEqual(y, 0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC